Question

If $I$ be the set of integers and if the relation R be defined over $I$ by $aRb$ iff $a-b$ is an even integer, $a,b\in I$, the relation is

• A. reflexive
• B. anti-symmetric
• C. symmetric
• D. equivalence

Answer 1

Answer: (A), (C), (D)

The correct options are
A reflexive
C symmetric
D equivalence

(1) Reflexivity : Since $0=a-a$ is even.

Hence$aRa\forall a\in I.$

Therefore $R$ is reflexive over $I.$

(2)Symmetry : Since $a-b$ is even then $b-a=-(a-b)$ is also even and hence $aRb\Rightarrow bRa.$

Therefore $R$ is symmetric in $I$.

(3) Transitive : $aRb$ and $bRc$

$\Rightarrow a-b$ and $b-c$ are even number

$\Rightarrow (a-b)+(b-c)$ is even number

$\Rightarrow a-c$ is even

$\therefore aRb$ and $bRc\Rightarrow aRc$

Hence $R$ is an equivalence relation.